Can Shinigami die this way (Death Note)...?

Alice

New member
**This question contains spoilers**

I know they can die if they save a human from dying by killing another human, but what if they THINK they are saving that human but it really wouldn't have made a difference?
I'm trying to connect this to how Light convinced Rem she had to kill L in order to save Misa's life.
At first in the beginning of the series, Rem said she'd kill L and it wouldn't matter since she is not saving anyone and he is just another human that will extend her life, but then in the infamous episode 25, Rem kills L and then dies.
But at the moment, Misa was not in any danger (unlike when she was being attacked and the shinigami Jealous saved her) she also had been allowed to leave so she was no longer a suspect.
I'm wondering why did Rem die after killing L... in order for a shinigami to die do they need to at least THINK they sacrificed themselves for a human?
Sorry if I'm missing something obvious, but I want to be sure about this.
 
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